What is Tarski’s definition of real number multiplication?

Question

Alfred Tarski came up with the following axiomatization of the real numbers, which only references the notions of “less than” and addition:

1. If $x < y$, then not $y < x$. That is, “$<$" is an asymmetric relation.
2. If $x < z$, there exists a $y$ such that $x < y$ and $y < z$. In other words, "$<$" is dense in $\mathbb{R}$.
3. "$<$" is Dedekind-complete. More formally, for all $X,Y \subseteq \mathbb{R}$, if for all $x \in X$ and $y \in Y$, $x < y$, then there exists a $z$ such that for all $x \in X$ and $y \in Y$, if $z \neq x$ and $z \neq y$, then $x < z$ and $z < y$.
4. $x + (y + z) = (x + z) + y$.
5. For all $x$, $y$, there exists a $z$ such that $x + z = y$.
6. If $x + y < z + w$, then $x < z$ or $y < w$.
7. $1\in\mathbb{R}$
8. $1 < 1 + 1$.

But it’s still equivalent to the usual axiomatization of the real numbers, which includes axioms for multiplication. Here is what Wikipedia says:

Tarski sketched the (nontrivial) proof of how these axioms and primitives imply the existence of a binary operation called multiplication and having the expected properties, so that $\mathbb{R}$ is a complete ordered field under addition and multiplication. This proof builds crucially on the integers with addition being an abelian group and has its origins in Eudoxus' definition of magnitude.

My question is, what is Tarski’s definition of multiplication in this system?

I skimmed Tarski’s book “Introduction to Logic and to the Methodology of Deductive Sciences”, and I found the above axioms, but I couldn’t find a definition of multiplication or a proof that multiplication satisfies the usual properties.

Answer 1

The results of our searching for Tarksi's definition of multiplication is given in the next section. In this section we 'cut-to-the-chase', sketching how to apply the Eudoxus theory of ratios.

Like Eudoxus/Euclid and other ancients, in this exposition numbers will always be positive; we are working in $(\Bbb R^{\gt 0}, 1, +)$. Before starting note that $(\Bbb N^{\gt 0}, 1, +)$ is naturally included in $(\Bbb R^{\gt 0}, 1, +)$.

We define the ratio of $u,v \in \Bbb R^{\gt 0}$ as a binary relation in $\Bbb N^{\gt 0} \times \Bbb N^{\gt 0} $,

$$\quad u \mathbin{:} v = \{ (n, m) : nu > mv\}$$

where $nu$ and $mv$ represent repeated addition. So the ancients could work with real numbers via ratios without a decimal system.

As a sanity check, the power set of $\Bbb N^{\gt 0} \times \Bbb N^{\gt 0} $ has the power of the continuum.

We only state what we need here from the ancient theory of proportions
(c.f. Euclid's Elements.Book V.Proposition 14).

Theorem: For any $x, y \in \Bbb R^{\gt 0}$ there exist one and only one number $z \in \Bbb R^{\gt 0}$ satisfying the following

$\quad \text{For every } m, n \in \Bbb N^{\gt 0}$
$\quad \quad \quad \quad \quad \quad [$ $\tag 1 nx \lt m \; \text{ iff } \; nz \lt my$ $\quad \quad \quad \quad \quad \quad \text{and}$
$\tag 2 nx = m \; \text{ iff } \; nz = my$ $\quad \quad \quad \quad \quad \quad \text{and}$
$\tag 3 nx \gt m \; \text{ iff } \; nz \gt my$ $\quad \quad \quad \quad \quad \quad ]$

You can think of above result as a variation of the 'squeeze theorem' by letting $n \to +\infty$ and taking the largest $m$ such that

$$\quad \frac{m}{n} \le x \le \frac{m+1}{n} \; \text{ and } \; \frac{m}{n}y \le z \le \frac{m+1}{n}y$$

is true.

Definition: For any $x, y \in \Bbb R^{\gt 0}$ the number $z$ from the theorem is denoted by $x \times y$. The corresponding binary operation on $\Bbb R^{\gt 0} \times \Bbb R^{\gt 0}$ is called multiplication.

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I found an online version of Tarski's book.

The book DOES NOT define multiplication!

In the last chapter of the book, chapter 10, two axioms systems for the real numbers are presented in a 'survey' fashion,

$\mathcal A'$ (the one where the OP lists the axioms) and in summary Tarski writes

System $\mathcal A'$ expresses the fact that the set of all numbers is a densely and continuously ordered Abelian group with respect to the relation < and the operation of addition, and it singles out a certain positive element 1 in that set.

and

$\mathcal A''$, and in summary Tarski writes

System $\mathcal A''$ expresses the fact that the set of all numbers is a continuously ordered field with respect to the relation < and the operations of addition and multiplication, and singles out two distinct elements 0 and 1 in that set, of which the first is the identity element for addition, and the second, the identity element for multiplication.

In Section 62
$\quad$Closer characterization of the first axiom system;
$\quad$its methodological advantages and didactic disadvantages

Tarski writes that

Even constructing the definition of multiplication and deriving the basic laws for this operation are not easy tasks to carry through.

Later in Section 65

$\quad$Equipollence of the two axiom systems;
$\quad$methodological disadvantages and didactic advantages of the second system

Tarski writes that

...both the definition of multiplication on the basis of the first system, and the subsequent proofs of the basic laws governing this operation, present considerable difficulties (while these laws appear as axioms in the second system).

Now wikipedia writes, in regard to $\mathcal A'$,

Tarski sketched the (nontrivial) proof of how these axioms and primitives imply the existence of a binary operation called multiplication and having the expected properties, so that R is a complete ordered field under addition and multiplication.

But that sketch is not to be found in the one (relevant) reference wikipedia gives - the book the OP is reading!

Wikipedia also states

Tarski proved these 8 axioms and 4 primitive notions independent.

And again, no reference.

Besides the primitive terms, $\mathcal A''$, with multiplication 'built-in' contains 20 axioms.

The last thing you will find in the book (besides the index) are the exercises for Chapter 10 and the last exercise is

*22. Derive all the axioms of System $\mathcal A'$ from the axioms of System $\mathcal A''$.

Tarski's book doesn't have any references.

The OP might find the link

Talk:Tarski's axiomatization of the reals

of interest. Apparently some mathematicians are trying to come up with the definition of multiplication in System $\mathcal A'$, and one came up with

Answer 2

I don't have access to Tarski's exposition, but the following arguments (see Sections 1-3 below) are all made in the same 'playground' that Tarski developed his theory.

I have no doubt that Tarski's definition of multiplication of the reals depends on using the Eudoxus Theory of Proportion (see this). The Eudoxus theory can be used to show that any two endomorphisms on the additive group of positive real numbers under addition commute (under functional composition), and that is crucial to defining multiplication with endomorphisms in our sketched-out theory.

Here is Definition 5 of Euclid's Book V:

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

Also from the wikipedia link,

The Eudoxian definition of proportionality uses the quantifier, "for every ..." to harness the infinite and the infinitesimal, just as do the modern epsilon-delta definitions of limit and continuity.

I can't say exactly how Tarksi defines multiplication, but I'm about 99% confident in the following:

There is one and only one binary operation of multiplication defined over $(\Bbb R, 0, 1, +, \le )$ satisfying

$\quad$ $1 \times 1 = 1$

$\quad$ Multiplication is a commutative and associative operation

$\quad$ Multiplication distributes over addition

$\quad$ If $0 \lt a \lt b$ and $c \gt 0$ then $0 \lt ca \lt cb$

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Section 1

With Tarski's axioms we start with

$$ (\Bbb R, 0, 1, +, \le ) \quad \text{ the additive group of numbers on the line (extending in both directions)}$$

There is no multiplication but $1 \gt 0$ is selected as the unit of measure.

The ancient Greeks, Eudoxus/Euclid/et.al, worked with $(\Bbb R^{>0},1,+)$ as a system of magnitudes. In the next section, we state three theorems, using modern mathematical terminology, where some of their logic is employed. Theorem 3 is an immediate consequence of the first two theorems.

In the last section we use this theory to define multiplication on $\Bbb R$, by again stating theorems without proof.

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Section 2

Theorem 1: Every endomorphism $\phi: \Bbb R^{>0} \to \Bbb R^{>0}$ is completely determined by knowing the image under $\phi$ of $1$. Each of these endomorphisms,

$$\tag 1 \phi_m:1 \mapsto m$$

is a bijective transformation, and so, the inverse ${\phi_m}^{-1}$ can also be recast into a $\text{(1)}$ representation. Finally, to any $m \in \Bbb R^{>0}$ there corresponds a $\text{(1)-form }\phi_m$.

This group is denoted by $\mathcal G$.

Theorem 2: The group $(\mathcal G, \circ)$ is commutative.

Theorem 3: Corresponding to any choice of $1 \in (\Bbb R^{>0},+)$ the group $\mathcal G$ of endomrophisms can be put in a bijectice correspondence with $\Bbb R^{>0}$. In this way a commutative binary operation,

$$\tag 2 x \times y = [\phi_x \circ \phi_y]\, (1) = \phi_x(y) = \phi_y(x)$$

call it multiplication of $x$ with $y$, $xy$, can be defined on $\Bbb R^{>0}$. This operation distributes over addition

$$\tag 3 x(y+z) = xy + xz$$

has a multiplicative identity

$$\tag 4 1x = x1 = x$$

and associated with every $x \in \Bbb R^{>0}$ is a number unique $y \in \Bbb R^{>0}$ such that

$$\tag 5 xy = yx = 1$$

Recall that we can write $y = x^{-1}$ or $x = y^{-1}$ when $\text{(4)}$ is true.

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Section 3

Proposition 4: Every endomorphism $\phi_m$ in $(\Bbb R^{>0},1,+)$ has one and only one extension to a (bijective) endomorhism on the abelian group $(\Bbb R,0,1,+)$. The collection $\mathcal P$ of these transformations forms a commutative group isomorphic to $\mathcal G$.

Recall that we have the inversion endomorphism $\gamma: x \mapsto -x$ defined on the commutative group $(\Bbb R,0,1,+)$.

Proposition 5: The inversion mapping $\gamma$ commutes with every endomorphism in $\mathcal P$.

Recall that we have the constant trivial endomorphism $\psi_0: x \mapsto 0$ defined on $(\Bbb R,0,1,+)$; it commutes with every other endomorphism on $(\Bbb R,0,1,+)$, and in particular every morphism in $\mathcal P$.

Proposition 6: The expression

$$\tag 6 \mathcal A = \mathcal P \cup \{\gamma \circ \phi_m \, | \, \phi_m \in \mathcal P \} \cup \{\psi_0\}$$

represents a disjoint union of endomorphisms on $(\Bbb R,1,+)$.

Proposition 7: The set $\mathcal A$ is closed under the operation of functional composition and this operation is commutative. Every endomorphism $\phi: \Bbb R \to \Bbb R$ belonging $\mathcal A$ is completely determined by knowing the image under $\phi$ of $1$. Except for the trivial $0\text{-endomorphism}$, each of these these mappings,

$$\tag 7 \phi_m:1 \mapsto m$$

is a bijective transformation with its inverse also belonging to $(\mathcal A,\circ)$.
Finally, to any $m \in \Bbb R$ there corresponds a $\text{(7)-form }\phi_m$.

So the trivial endomorphism $\psi_0$ on $\Bbb R$ can be written as $\phi_0$ and we can also write

$$\tag 8 \mathcal A = \{ \phi_m \, | \, m \in \Bbb R\}$$

Theorem 8: The structure $(\Bbb R,0,1,+)$ can be put into a $1:1$ correspondence with $\mathcal A$. In this way a second binary operation, multiplication, can be defined over $(\Bbb R,0,1,+)$. The new algebraic structure, $(\Bbb R,0,1,+,\times)$, forms a field.

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Note: An outline for some of the above theory can be found in this article,

$\quad$ Translating Tarski's Axiomatization/Logic of $\mathbb R$ to the Theory of Magnitudes